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"It has long been an axiom of mine that the little things are infinitely the most important" (Sherlock Holmes)

Just a note

• The first function (#1.) simply calculates the X2 statistic from the frequencies in variable y, and assumes y=a, y=b, y=c, y=d - where a,b,c,d, are the four cell frequencies in a 2×2 table. Note, the value of X2 is not available (NA) if any of the margin totals (a+b, or c+d, or a+c, or b+d) equal zero.

This function is needed for functions #3 and #4 to work.

• The second function (#2.) is a random independent binomial function, and is needed in order to estimate the distribution of the X2 statistic assuming you have 2 samples, each of which represent binomially distributed populations.

• The third function (#3.) takes R (=5000) random samples from a multinomial distribution, whose expected frequencies are calculated from the margin totals of your sample - assuming the null hypothesis to be true. The X2 statistic of each of these (bootstrap) samples is calculated, and arranged in ascending order. By finding what proportion of this bootstrap distribution equals or exceeds the observed value of X2 a (conventional or mid-P) 1-tailed P-value is returned.

• The fourth function (#4.) does exactly the same thing, but uses the independent binomial function, and assumes that a+b=n1 and c+d=n2.